Weak Inhomogeneity Expansion in Polymer Field Theory

BY Yi-Xin Liu


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This report is originally an internal document of Yi-Xin Liu’s group, which is generated at Jan. 25, 2016. The PDF version can be downloaded via the following link. Comments are welcome!

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$ \newcommand\vx{\mathbf{x}} \newcommand{\der}[2]{\frac{\partial {#1}}{\partial {#2}}} \newcommand{\abs}[1]{\left\lvert {#1} \right\rvert} \newcommand{\vk}{\mathbf{k}} \newcommand{\vzero}{\mathbf{0}} \newcommand{\Der}[2]{\frac{\delta {#1}}{\delta {#2}}} \newcommand{\lap}{\nabla^2} $

Abstract

This report performs weak inhomogeneity expansion for the continuous Gaussian chain model of a homopolymer in an external field. The aim is to understand where does the Debye function come from. This technique constitutes the main parts of the random phase approximation (RPA). By extending this technique from single-chain formulation to many-chain formulation and regarding the potential field arising from interaction with other chains in the same system as an external field, RPA is equivalent to the weak inhomogeneity expansion.

Continuous Gaussian Chain

The normalized single partition function for a continuous Gaussian chain under an external field is given by1

where the physical length is scaled by the radius of gyration of a non-perturb Gaussian chain $R_g=\sqrt{Nb^2/6}$. The volume $V$ is also dimensionless scaled by $R_g^3$.

The propagator $q$ in the above equation satisfies the Fokker-Planck equation, also known as the modified diffusion equation (MDE)

subject to the initial condition

Note that to write the MDE in the form as in eq. \eqref{eq:MDE}, $w(\vx)$ is actually $N$ times the external potential field. Assuming the external field is $W(\vx)$, then $w(\vx)=NW(\vx)$

For a given external field $w(\vx)$, we can obtain the propagator by solving eq. \eqref{eq:MDE}.

Weak Inhomogeneity Expansion

In general, it is impossible to find an analytic solution to the MDE with a general $w(\vx)$. However, a particularly perturbation expansion can be derived when the applied potential field $w(\vx)$ has inhomogeneities that are weak in amplitude. To define such a situation, we introduce the volume average of the potential

and re-express $w(\vx)$ according to

which serves to define the inhomogeneous part of the field, $\epsilon\omega(\vx)$. For weak inhomogeneities, a small parameter $\epsilon$ ($\abs{\epsilon}\ll 1$) describes their characteristic amplitude. For the continuous Gaussian chain model of a homopolymer, the MDE and initial condition become

where the functional dependence of q on $w(\vx)$ has been suppressed in our notation. The term proportional to $w_0$ on the right-hand side of eq.\eqref{eq:MDE-w0} can be removed by the substitution

which leads to

A weak inhomogeneity expansion can be developed by assuming that $p(\vx, s)$ can be expressed as

where the $p^{(j)}(\vx, s)$ are independent of $\epsilon$. In eq.\eqref{eq:p-expansion} we adopt the conventional notation $\sim$ to indicate an asymptotic expansion. As such, the infinite series on the right-hand side may be either convergent or divergent. Even when it does not converge, eq.\eqref{eq:p-expansion} can still be useful in truncated form for approximating $p(\vx, s)$ at sufficiently small $\epsilon$.

The $p^{(j)}$ are calculated by inserting eq.\eqref{eq:p-expansion} into eq.\eqref{eq:MDE-omega} and equating terms order by order in $\epsilon$.

Zero-th Order Solution

At leading order, $O(\epsilon^0)$, we have

which has the trivial solution

First Order Solution

At $O(\epsilon^1)$, the corresponding equations (see Appendix A for derivation) are

Provided the system under consideration is unbounded or subject to periodic boundary conditions, this initial value problem is most easily solved by means of spatial Fourier transforms. Defining Fourier transforms in accordance with

and assuming that the Fourier transform of $\omega(\vx)$ exists, denoted by $\hat{\omega}(\vk)$, one finds that (See Appendix B for derivation)

where the carets denote Fourier-transformed quantities and

Second Order Solution

At $O(\epsilon^2)$, the corresponding equations (see Appendix C for derivation) are

A similar procedure leads to (see Appendix D for derivation)

with

Expansion of $Q$

We can expand the normalized single partition function $Q$ based on above perturbation expansion of the propagator $q$. The propagator being expanded to $O(\epsilon^2)$ is

Therefore we can expand $Q$ as

It is more convenient to write the integrals of $p^{(j)}$ in Fourier space, which can be expressed as

and

Now we can express $Q$ in the Fourier space as

The expansion of $Q$ can be further simplified. Firstly, we know that the volume average of the fluctuation of the potential field is 0 because

The value of the Fourier transform of the potential field at zero wave number is equal to this average because

Therefore, one obtains

Substituting this into eq.\eqref{eq:p1} gives

Secondly, from eq.\eqref{eq:p2} we know

In the second line, we only change the summation variable from $\vk’$ to $\vk$. Noting that

Let

Then it becomes

This is where the well known Debye function comes from, which is defined as

Therefore,

Insert eq.\eqref{eq:p101-Fourier}, \eqref{eq:p201-h3}, and \eqref{eq:h301-final} into eq.\eqref{eq:Q-expan-v2}, we arrive the final expansion of $Q$ in the Fourier space

or, by inverting the Fourier transforms,

If we define the inverse transform of the Debye function as

Then the expansion of $Q$ in real space can be written as

Expansion of the Density Operator

A weak inhomogeneity expansion for the segment density operator $\rho(\vx; [w])$ can be obtained in one of two equivalent ways. One way is expressing $\rho(\vx; [w])$ as an integral of the propagator, such as

and substituting the weak inhomogeneity expansion of the propagator eq.\eqref{eq:q-p} and \eqref{eq:p-expansion} into above equation and keeping to a certain order. The other way is performing a direct functional differentiation of eq.\eqref{eq:Q-expan-final} according to

Here we Follow the latter approach. From eq.\eqref{eq:Q-expan-final} we have

From eq.\eqref{eq:w0-w} we have

From eq.\eqref{eq:w-omega} we have

Now we can perform the functional differentiation as

where $\rho_0 \equiv N/V$ is the volume-average segment density of a single chain. If we only retain the first order contribution of $\epsilon$, above equation can be simplified to

Or we can use eq.\eqref{eq:w-omega} to rewrite $\rho$ as a functional of $w$ instead of $\omega$

We can also inverse the above equation to predict what external field should be applied if we want to obtain a certain segment density. Such inversion can be done in the Fourier space

where $\phi=\rho/\rho_0 - 1$ is the dimensionless fluctuation of segment density around its volume-average value and $\Delta w=w-w_0$ is the fluctuation of the external field. It is straightforward to inverse it

Appendix

A. Derivation of eq.\eqref{eq:MDE-p1}

At $O(\epsilon^1)$, the expansion for $p(\vx, s)$ is

Inserting it into eq.\eqref{eq:MDE-omega}, we have

From the zeroth order expansion eq.\eqref{eq:MDE-p0}, we know

It simplifies eq.\eqref{eq:MDE-full-A} to

As $\epsilon \to 0$, the last term in the right-hand side of the above equation vanishes, leading to eq.\eqref{eq:MDE-p1}.

B. Derivation of eq.\eqref{eq:p1}

We know

which is the trivial solution of zeroth order expansion. Therefore eq.\eqref{eq:MDE-p1} becomes

Perform Fourier transforms on both sides of above equation, we reaches

Re-organize above equation into the form

It can be easily solved and the solution is

Let $s=0$, and with $\hat{p}^{(1)}(\vk, 0)=0$ (because $p^{(1)}(\vx, 0)=0$), we then have

or

Write eq.\eqref{eq:p1-intermediate-A} with $\hat{p}^{(1)}$ in the left-hand side and substitute above equation into it, we arrives at

which is equivalent to eq.\eqref{eq:p1}.

C. Derivation of eq.\eqref{eq:MDE-p2}

At $O(\epsilon^2)$, the expansion for $p(\vx, s)$ is

Inserting it into eq.\eqref{eq:MDE-omega}, we have

Substitute eq.\eqref{eq:MDE-p0} and \eqref{eq:MDE-p1} into above equation and ignore terms with $\epsilon$, it simplifies to eq.\eqref{eq:MDE-p2}.

D. Derivation of eq.\eqref{eq:p2}

We copy eq.\eqref{eq:MDE-p2} here

Perform Fourier transforms on both sides of above equation, we arrives

This is a general first order linear partial differential equation with respect to $s$

The solution is

where $C$ is a constant which can be obtained by applying the initial condition, and

From the first order solution, we can find

Substituting eq.\eqref{eq:p1} into above equation, we have

To simplify the notation, we define

Substitute eq.\eqref{eq:us-A} and \eqref{eq:wp1-A} into eq.\eqref{eq:p2-raw-A}

Because $p^{(2)}(\vx, 0)=0$, thus $\hat{p}^{(2)}(\vk, 0)=0$, therefore

Thus

Substitute it back into previous equation,

We can define

which completes the derivation.

Acknowledgements

This note is supported by the by the China Scholarship Council (No. 201406105018).

References

  1. Fredrickson, G. H. The Equilibrium Theory of Inhomogeneous Polymers; Clarendon Press: Oxford, 2006. 

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